## On the order of the Maass $$L$$-function on the critical line.(English)Zbl 0724.11029

Number theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 325-354 (1990).
Let $$\Psi$$ be a Maass form for the full modular group with eigenvalue $$1/4+\kappa^2$$. Suppose that $$\Psi$$ is a Hecke eigenform, normalized by having $$\sqrt{y}K_{i\kappa}(2\pi y)e^{2\pi ix}$$ as its Fourier term of order 1. Its $$L$$-function $$L_{\Psi}$$ is shown to satisfy $L_{\Psi}(1/2+it) \ll_{\varepsilon} \| \Psi \| \sqrt{\kappa} e^{\pi \kappa /2} t^{1/3+\varepsilon}$ for $$t\gg 1$$, $$\kappa \ll t^{1/3}$$ and $$\varepsilon >0$$, the $$L^2$$-norm of $$\Psi$$ is denoted by $$\| \Psi \|$$. This result improves the estimate $$O(t^{1/2+\varepsilon})$$ in [C. Epstein, J. L. Hafner and P. Sarnak, Math. Z. 190, 113–128 (1985; Zbl 0565.10026)]. The proof is considerably more complicated.
The central point in the proof is the estimation of $$\int^{2N}_{M=N}| \sum^{2M}_{n\geq M}\rho (n)n^{-s}| \,dM;$$ the $$\rho(n)$$ are Fourier coefficients of $$\Psi$$. The sum over $$n$$ is split up into many small subsums. The subsums are treated by a method of M. Jutila [Lect. Notes Math. 1380, 120–136 (1989; Zbl 0674.10032)].
[For the entire collection see Zbl 0694.00005.]

### MSC:

 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F37 Forms of half-integer weight; nonholomorphic modular forms 11M41 Other Dirichlet series and zeta functions

### Keywords:

method of Jutila; critical line; Maass form; L-function; estimation

### Citations:

Zbl 0694.00005; Zbl 0565.10026; Zbl 0674.10032